Pupil Text MEP: Codes and Ciphers, UNIT 20 Enigma Cipher 20 Enigma Cipher Introduction Security blunders on both sides during the First World War increased the need for a higher level of secrecy and more advanced methods of enciphering messages other than traditional pencil and paper techniques. In 1915, two Dutch Naval officers invented a machine to encrypt messages. This encryption tool became one of the most notorious of all time, the Enigma cipher machine. Arthur Scherbius, a German businessman, patented the Enigma in 1918 and began selling it commercially to banks and businesses. The Enigma machine's place in history was secured in 1926 when the German armed forces began using a specially adapted military version to encrypt their communications. They continued to rely on the machine throughout World War II, believing it to be absolutely unbreakable. How the Enigma machine worked When a plaintext letter was typed on the keyboard, an electric current would pass through the different scrambling elements of the machine and light up a ciphertext letter on the 'lampboard'. What made the Enigma machine so special was the fact that every time a letter was pressed, the moveable parts of the machine would change position so that the next time the same letter was pressed, it would most likely be enciphered as something different. This meant that traditional frequency analysis methods could not be used to crack the code. Rotors Lampboard Plugboard (covered) Keyboard World War II Enigma Cipher Machine To make it even more difficult, different parts of the machine could be set up in different ways, with each setting producing a unique stream of enciphered letters. Unless you knew the exact setting of the machine, you couldn't decipher the messages. Choosing rotors Inside the Army issue Enigma machine there were 3 cipher wheels called rotors which could be taken out and changed about. Each rotor had the letters A to Z with different 1 Pupil Text MEP: Codes and Ciphers, UNIT 20 Enigma Cipher internal wiring systems. There were 5 rotors that the Enigma operator could choose from for the 3 slots in the machine. E D C B A Z Y X Diagram of an individual rotor Rotors in position in an Enigma Cipher Machine Exercise 1 How many ways are there of positioning the 5 rotors in the 3 slots in an Enigma machine? Starting position of rotors The rotor starting point could be any one of 26 different positions (one for each letter of the alphabet). Exercise 2 In how many different ways can you set the starting position of the rotors? Ring settings Each time a letter was pressed on the keyboard, the rotor on the far right would move around one place. At one particular position it would kick the middle rotor forward one position. Then after a further complete revolution it would again kick the middle rotor forward by one position. Likewise the middle rotor at one of its positions would kick the left hand rotor forward by one position. The system was similar (but not identical to) a milometer in a car except that a revolution would involve 26 steps forward rather than 10. (In fact the movements of the middle rotor were slightly more complex than has been described here.) Each rotor had a metal ring attached to its circumference marked with the letters A–Z. The rings could be moved round the inner cores of their rotors and then locked at chosen positions. These three chosen positions were known as the ring settings. The chosen settings did not affect the basic electrical configuration of the machine, but one thing they did was to fix the positions at which the forward movements of the middle and left rotor occurred. We are now in a position to determine the number of possible ways of setting up the rotors. It will be 60× 17576 ↓ ↓ No. of ways of No. of different starting positioning 5 rotors positions for rotors in 3 positions 2 Pupil Text MEP: Codes and Ciphers, UNIT 20 Enigma Cipher This gives 1 054 560. So already there are already over one million possible settings for the start up position of the machine – but it is even more complicated! Plugboard On the front of the machine was another variable section called the plugboard. This was used to further scramble the messages, and increase the possible number of ways the machine could be set up. A B C D E F G H I J K L M N O P Q R S T U V W X Y Z The Enigma machine had several cables with a plug at each end that could be used to plug pairs of letters together on the plugboard. If A were plugged to B then upon typing the letter A, the electric current would follow the path through the machine that was normally associated with the letter B, and vice versa. Example If there was just one cable, in how many different ways could the plugboard be set up? Solution You could connect A with 25 other letters B with 24 other letters C with 23 other letters, etc. giving 25+++++ 24 23... 2 1 ways. You can easily see this as =++++ S25 25 24... 2 1 =++++ S25 1 2... 24 25 =++++=× 2S25 12 2626 4444 ... 4444 26262526 3 25 times 25× 26 nn( + 1) S= or S = 25 2 n 2 = 325 ways 3 Pupil Text MEP: Codes and Ciphers, UNIT 20 Enigma Cipher To find out the number of connections to use that gives the maximum number of possibilities requires results from combinatories. We will start by working through the following Activity. Activity 1 A B C With this simplified plugboard, determine how many ways there are of making connections with a) 1 b) 2 c) 3 cables D E F What is the optimum number to use? Activity 1 shows how complicated it can get even with just 6 letters. In fact, the key formula for the number of ways of choosing m pairs out of n objects (n must be an even number) is n! (nmm−22)!!m (A proof is given in Appendix 1.) Example In how many ways can you choose 2 pairs from 6 objects? Solution Here nm==62, , so 6! number of ways = = 45 222!!! (and this is what you should have obtained in Activity 1). Exercise 3 Check the other two values found in Activity 1. In fact, with 26 letters the Germans used 10 cables, i.e. 10 pairs. Example How many ways are there of choosing 10 pairs from 26 letters? Solution nm==26, 10, so from the formula 26! number of ways = 6102!!10 =150 738 274 937 250 ≈×15. 1014 4 Pupil Text MEP: Codes and Ciphers, UNIT 20 Enigma Cipher Activity 2 Write a program or use a spreadsheet to evaluate the formula n! (nmm−22)!!m for nm==26 and 1 , 2, ..., 13 Does the result surprise you? Finally we can work out the approximate number of ways of setting up the electrical circuits on the three rotor Enigma machine; it is 60×××=× 17576 1.. 5 1014 1 58 10 20 ↓ ↓ No. of set-up No. of ways of positions for rotors setting up plugboard which is a very large number! Deciphering Enigma The process of deciphering was incredibly simple, provided the recipient of the message knew how the Enigma machine had been set up when the message had been enciphered. A German soldier receiving an enciphered message would simply have to type the ciphertext letters into their own Enigma machine. If their machine was set up exactly in exactly the same way as the message sender's, the plaintext letters would appear on the lampboard. In this way, the algorithm, or method of encryption, is the Enigma machine. The key is knowing how the machine is set up. This type of encryption is known as Symmetric Encryption because the operation of deciphering is inverse to the operation of deciphering. The decoding key is also the same as the encoding key. This means that if the enemy knows your method of encryption (and during World War II, the Allies knew the Germans were using Enigma), then the key must be kept secret. If the enemy uncover the key, this immediately implies cracking of the message thus jeopardising security. In order to make it as difficult as possible for the Allies to work out the Enigma key, the Germans would change the key every day, resetting their Enigma machines at midnight every night. Cipher machine operators were issued with a key sheet every month, which told them how to set up their Enigma machines for every day that month. There was an obvious security weakness in this system in that if the Allies were able to recover the key sheets, they would be able to read the Enigma messages. For this reason, key sheets were extremely closely guarded and were printed in soluble ink. If it ever looked as though the key sheet might be captured by the Allies, German soldiers would dip the key sheet in water, and wash off all the information. The Germans believed the strength of the Enigma lay in the fact that it was impossible to work out the key from the billions and billions of potential keys every single day. As long as the Allies did not get hold of the key sheet, they thought that their communications would remain secure. 5 Pupil Text MEP: Codes and Ciphers, UNIT 20 Enigma Cipher Whilst the methods used at Bletchley Park are too complicated to fully illustrate here, we can gain some insight into the process by using a 'Paper Enigma'.